In the most basic sense, a minimal surface is one that can locally minimize its surface area. Such a surface has a mean curvature of zero! Curvature is how much a surface, or parameterized curve changes at each point. Consider the sine curve. Intuitively it can be understood that it is "curvier" than a line, or even a parabola. Mean curvature is the average of the sum of curvature at each point.
Rigorously parameterizing a minimal surface poses a formidable challenge, and a relatively small amount of these surfaces have been discovered. The simplest, but most redundant example is the plane. It has no curvature at any point, so of course it has a mean curvature of zero across the whole surface!
Shown below is a helicoid that encloses a soap film. This was taken from user "Blinking Spirit" on Wikipedia. The soap film forms a minimal surface defined between the helicoid. The reason for this is described later on.
The parameterized minimal surfaces take on exotic forms since they must fit such strict criteria. The criteria that they must have a mean curvature of zero is just one of several criteria that a minimal surface must meet. The helicoid is a familiar example of a minimal surface. A famous, but exotic looking example is the saddle tower. Wikipedia user "Anders Sandberg" created the image below of the saddle tower. Despite its odd shape, it has a mean curvature of zero!
As stated above, a soap film can form a minimal surface between two arbitrary curves. This should not be surprising though, because the soap film must achieve the lowest possible energy between the two curves. The energy of the film is the product of the tension the surface experiences and the surface area. To survive as film, it must decrease the surface area as much as possible because tension will more or less stay constant. In doing so, it achieves the lowest possible potential energy, which allows it to persist as a film.
So despite rigorously parameterizing a minimal surface, they can be easily be created in a physical sense through creating a soap film between two completely arbitrary curves! To illustrate this, I created the following objects.
In a simple example, I created two non-concentric circles of variable radii, shown below.
The lower circle has a diameter of 3 inches when printed at an appropriate scale while the upper circle has a diameter of 2 inches. They are held together by a cylinder which has a height of roughly 6 inches. This will not only hold the circles together, but allow for them to be dipped in the soap solution. The soap film will extend from the bottom curve to the top. However, if the potential energy of the surface is too high, then it will break and instead form inside each respective curve rather than between them!
My second object aims to prove that a soap film can create a minimal surface between two completely arbitrary curves. I once again created two non-concentric circles but this time made their height vary by the product of sine and cosine. Their radii are 1.5 inches when printed at scale and 1.2 inches respectively and the cylinder between them will have a height of 6 inches.
My last object aims to facetiously reproduce a result described in a review of the computational study of organic semiconductors by Coropceanu et al. in 2007 from their work Charge Transport in Organic Semiconductors. They showed that the transfer integral (how well molecular orbitals overlap to facilitate charge transfer) decreases as two cofacial tetracene molecules move away from each other in space. We know that the soap film has less likelihood of surviving as two curves move apart, so obviously, this is a very good representation of the transfer integral, with no false equivalencies whatsoever.
Likewise, the transfer integral is variable as two cofacial tetracene molecules slide past each other in space, while at a fixed horizontal distance between each other. Once again, this is definitely best modeled through soap! What else could do this?
My object, which is definitely suitable for publication in Materials Horizons, is shown below. The real length of the object is 6 inches, but will be scaled down to 3 inches. The fused hexagons are 2 inches apart, and the cylinder which holds them together will have a scaled height of roughly 4 inches.
Comments
Post a Comment